A detailed view of the Gaussian–Lorentzian sum and product functions and their comparison with the Voigt function

Abstract

The Gaussian–Lorentzian sum (GLS) and product (GLP) functions remain important in X‐ray photoelectron spectroscopy (XPS) peak fitting. Here, we present a detailed view of these functions, comparing them with each other and with the Voigt function (the “LA(m)” function). First, we show the GLS, GLP, and LA(m) functions as a function of their mixing parameters, m, which reveals differences between them. We then illustrate the use of these functions to fit a series of spectra acquired at different pass energies (resolutions). Next, we show the underlying Gaussian and Lorentzian components of a series of GLS and GLP functions as a function of m, which confirms that the GLS is a simple linear combination of Gaussian and Lorentzian functions. However, one of the two functions used to make the GLP can be very wide, that is, at its extremes, one of these functions has infinite width. We then discuss a plot of the areas of the GLS, GLP, and LA(m) functions as a function of m, which reveals the expected, linear increase in area of the GLS, but nonlinear changes in the areas of the other two functions. Finally, to better understand them, we fit these functions to each other. These results indicate that the GLS and GLP better match the LA(m) function at lower and higher values of the mixing parameter, respectively.

Implication of Current–Voltage Curve Shape in Molecular Electronics

“…33,34 The Voigt profile is often considered to be complex due to the mathematically expensive convolution procedure. 39 Hence, a simple sum (Gaussian−Lorentzian sum, denoted as GLS) or product (Gaussian−Lorentzian product, GLP) of Gaussian and Lorentzian functions is also widely used for line shape fitting in spectroscopy: 39−41 A striking feature in the Lorentzian function, compared to the Gaussian function, is a long tail (Lorentzian tail; Figure 2a). Due to this tail, a transmission peak can be overlapped with a bias window even at a low bias.…”

“…Broadening of the line shape in spectroscopy is addressed by the Voigt profile, which arises from the convolution of Lorentzian and Gaussian functions. , The Voigt profile is often considered to be complex due to the mathematically expensive convolution procedure . Hence, a simple sum (Gaussian–Lorentzian sum, denoted as GLS) or product (Gaussian–Lorentzian product, GLP) of Gaussian and Lorentzian functions is also widely used for line shape fitting in spectroscopy: − A striking feature in the Lorentzian function, compared to the Gaussian function, is a long tail (Lorentzian tail; Figure a). Due to this tail, a transmission peak can be overlapped with a bias window even at a low bias.…”

Guide to XPS data analysis: Applying appropriate constraints to synthetic peaks in XPS peak fitting

“…Nevertheless, these functions, and those derived from them, often differ significantly at their edges/wings. [3][4][5][6] The successful use of bell-shaped curves in XPS to interpret and model the specific chemical states of a material depends on two overarching concepts. The first is the extent to which signals from different chemical states overlap.…”

“…In both cases, the relative contributions of the two underlying functions is controlled by a mixing parameter, m, as indicated by SGL(m) or GL(m), where m takes on values of 0 -100, or equivalently (with minor adjustments to the mathematics) from 0 -1. [3][4][5][6] Nevertheless, a given value of m does not mean the same thing in these pseudo-Voigt (or Voigt) functions. That is, the GL(30) and SGL(30) functions are somewhat different, especially at their wings, and they are also different from a comparable Voigt function.…”

Surface analysis insight note: Synthetic line shapes, integration regions and relative sensitivity factors

“…The problem posed by measuring emission via components in a peak model compared to measuring emission by integrating data over an energy interval is that the latter is by definition over a finite interval, whereas in the case of the former, synthetic line shapes derived from Gaussian and Lorentzian functions 14,15 are defined in terms of the area over an infinite energy range. One might think that a solution is to truncate synthetic line shapes to the interval used to integrate intensity from data.…”